Bond Valuation. FINANCE 350 Global Financial Management. Professor Alon Brav Fuqua School of Business Duke University. Bond Valuation: An Overview

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1 Bond Valuation FINANCE 350 Global Financial Management Professor Alon Brav Fuqua School of Business Duke University 1 Bond Valuation: An Overview Bond Markets What are they? How big? How important? Valuation of bonds Zero-coupon bond Coupon bonds Interest rate sensitivity The term structure of interest rates 2

2 Definition of a Bond A bond is a security that obligates the issuer to make specified interest and principal payments to the holder on specified dates. Coupon rate Face value (or par) Maturity (or term) Bonds are also called fixed income securities. Bonds differ in several respects: Repayment type Issuer Maturity Security Priority in case of default 3 Types of Bonds: Repayment Pure Discount or Zero-Coupon Bonds Pay no coupons prior to maturity. Pay the bond s face value at maturity. Coupon Bonds Pay a stated coupon at periodic intervals prior to maturity. Pay the bond s face value at maturity. Floating-Rate Bonds Pay a variable coupon, reset periodically to a reference rate. Pay the bond s face value at maturity. Perpetual Bonds (Consols) No maturity date. Pay a stated coupon at periodic intervals. Annuity or Self-Amortizing Bonds Pay a regular fixed amount each payment period. Principal repaid over time rather than at maturity. 4

3 Types of Bonds: Issuers Bonds Government Bonds Mortgage-Backed Securities Municipal Bonds Corporate Bonds Asset-Back Securities Issuer US Treasury, Government Agencies Government agencies (GNMA etc) State and local government Corporations Corporations 5 U.S. Government Bonds Treasury Bills No coupons (zero coupon security) Face value paid at maturity Maturities up to one year Treasury Notes Coupons paid semiannually Face value paid at maturity Maturities from 2-10 years 6

4 U.S. Government Bonds Treasury Bonds Coupons paid semiannually Face value paid at maturity Maturities over 10 years The 30-year bond is called the long bond. Treasury Strips Zero-coupon bond Created by stripping the coupons and principal from Treasury bonds and notes. No default risk. Considered to be risk free. Exempt from state and local taxes. Sold regularly through a network of primary dealers. Traded regularly in the over-the-counter market. 7 Mortgage and Municipal Bonds Agencies Bonds: Mortgage-Backed Bonds Bonds issued by U.S. Government agencies that are backed by a pool of home mortgages. Self-amortizing bonds. (mostly monthly payments) Maturities up to 30 years. Prepayment risk. Municipal Bonds Maturities from one month to 40 years. Exempt from federal, state, and local taxes. Generally two types: Revenue bonds General Obligation bonds Riskier than U.S. Government bonds. 8

5 Corporate Bonds Bonds issued by corporations Bond indentures or covenants. Seniority: Secured bonds; Debentures. Fixed-rate versus floating-rate bonds. Investment-grade vs. below investment-grade bonds. Additional features: call provisions convertible bonds puttable bonds 9 Seniority of Corporate Bonds In case of default, different classes of bonds have different claim priority on the assets of a corporation. Secured Bonds (Asset-Backed) Secured by real property. Ownership of the property reverts to the bondholders upon default. Debentures Same priority as general creditors. Have priority over stockholders, but subordinate to secured debt. 10

6 Bond Ratings Moody s S&P Quality of Issue Aaa AAA Highest quality. Very small risk of default. Aa AA High quality. Small risk of default. A A High-Medium quality. Strong attributes, but potentially vulnerable. Baa BBB Medium quality. Currently adequate, but potentially unreliable. Ba BB Some speculative element. Long-run prospects questionable. B B Able to pay currently, but at risk of default in the future. Caa CCC Poor quality. Clear danger of default. Ca CC High speculative quality. May be in default. C C Lowest rated. Poor prospects of repayment. D - In default. 11 The US Bond Market: Amount ($bil.). Source: U.S. Federal Reserve (Table L.4, September/2006) Debt Instrument Treasury securities Municipal securities Corporate and foreign bonds 2006 Q2 4, , ,705.3 Consumer Credit Mortgages 2, ,757.7 Corporate equities Mutual fund shares 18, ,

7 U.S. Treasuries A Few Bond Markets Statistics U.S. Treasuries, May 20 th Bills MATURITY DISCOUNT/YIELD DISCOUNT/YIELD TIME DATE CHANGE 3-Month 08/16/ / / :41 6-Month 11/15/ / / :41 Notes/Bonds COUPON MATURITY CURRENT PRICE/YIELD TIME DATE PRICE/YIELD CHANGE 2-Year /30/ / / :08 3-Year /15/ / / :06 5-Year /30/ / / :07 10-Year /15/ / / :07 30-Year /15/ / / :07 13 Term Structure, May 20 th,

8 Bond Valuation: Zero Coupon Bonds B Market price of the Bond F Face value R Annual percentage rate m compounding period (typical: semiannual) i Effective periodic interest rate; i=r/m T Maturity (in years) N Number of compounding periods; N = T*m Two cash flows to purchaser of bond: B 0 at time 0 F at time T What is the price of a bond? B F = 1+ i Use present value formula: ( ) N 0 15 Valuing Zero Coupon Bonds What is the current market price of a U.S. Treasury strip that matures in exactly 5 years and has a face value of $1,000. The APR is R=7.5% (annual compounding)? 1, 000 = $ What is the APR on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $ (annual compounding)? 1,000 1, = R = 7 1 = 7.8% ( 1+ R) We also call R the yield to maturity. 16

9 Bond Prices and Interest Rates The case of zero coupon bonds Consider the following 1, 2 and 10- year zero-coupon bonds, all with face value of F=1000. APR of R=10%, compounded annually. We obtain the following table for increases and decreases of the interest rate by 1%: Interest Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year 9.0% $ $ $ % $ $ $ % $ $ $ Bond prices move up if interest rates drop, decrease if interest rates rise 17 Bond Prices and Interest Rates Bond prices are inversely related to interest rates Longer term bonds are more sensitive to interest rate changes than short term bonds The lower the IR, the more sensitive the price. 18

10 Measuring Interest Rate Sensitivity Zero Coupon Bonds We would like to measure the interest rate sensitivity of a bond or a portfolio of bonds. How much do bond prices change if interest rates change by a small amount Why is this important? Use Dollar value of a one basis point decrease (DV01): Basis point (bp): 1/100 of one percentage point =0.01%= Calculate DV01: Method 1: Difference of moving one basis point down: DV01= B(R-0.01)-B(R). Method 2: Difference of moving 1/2bp down minus 1/2pb up: DV01=B(R-0.005%) -B(R+0.005%). Method 3: Use calculus: db DV 01 = dr 1 10, Computing DV01: An Example Reconsider the 1, 2 and 10- year bonds discussed before: Interest Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year 9.990% $ $ $ % $ $ $ % $ $ $ % $ $ $ Method 1 $ $ $ Method 2 $ $ $ Method 3 $ $ $ db 1 dr 10,000 $1, , Method 3: = T = T *$0.10* T + 1 T

11 DV01: A Graphical Approach DV01 estimates the slope of the line on the Price-Interest rate curve. higher slope implies greater sensitivity 21 Prices of Coupon Bonds: Example 1: Amortization Bond Consider Amortization Bond T=2 m=2 C=$2,000 c = C/m = $2,000/2 = $1,000 R=10% i = R/m = 10%/2 = 5% How can we value this security? Brute force discounting Similar to another security we already know how to value? Replication 22

12 Prices of Coupon Bonds: Example 1: Amortization Bond Compare with a portfolio of zero coupon bonds: Coupon Bond Zero 1 Zero 2 Zero 3 Zero 4 Period\Price $3, $ $ $ $ $1, $1, $0.00 $0.00 $ $1, $0.00 $1, $0.00 $ $1, $0.00 $0.00 $1, $ $1, $0.00 $0.00 $0.00 $1, A First Look at Arbitrage Reconsider amortization bond; suppose bond trades at $3,500 Can make risk less profit Buy low: buy amortization bond Sell high: Sell portfolio of zero coupon bonds Coupon Bond Zero 1 Zero 2 Zero 3 Zero 4 Total Period\Price ($3,500.00) $ $ $ $ $ $1, ($1,000.00) $0.00 $0.00 $0.00 $ $1, $0.00 ($1,000.00) $0.00 $0.00 $ $1, $0.00 $0.00 ($1,000.00) $0.00 $ $1, $0.00 $0.00 $0.00 ($1,000.00) $0.00 riskless profit of $45.95 no riskless profit if price is correct 24

13 Valuation of Coupon Bonds: Example 2: Straight Bonds What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the interest rate is 10% compounded semiannually? Months B = + = $ Valuing Coupon Bonds The General Formula What is the market price of a U.S. Treasury bond that has an annual coupon C, face value F and matures exactly T years from today if the required rate of return is R, with m-periodic compounding? Semiannual coupon is: c = C/m Effective periodic interest rate is: i = R/m number of periods N = mt N c c c c c c+f B = [ Annuity]+ [ Zero]= c i i ( ) N + F 1+ i ( ) N 26

14 The Concept of a Yield to Maturity So far we have valued bonds by using a given interest rate, then discounted all payments to the bond. Prices are usually given from trade prices need to infer interest rate that has been used Definition: The yield to maturity is that interest rate that equates the present discounted value of all future payments to bondholders to the market price: Algebraic: B = c yield / m + ( 1+ yield / m) N ( 1 yield / m) N F 27 Yield to Maturity A Graphical Interpretation $2, $2, $1, $1, $ $0.00 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% Consider a U.S. Treasury bond that has a coupon rate of 10%, a face value of $1,000 and matures exactly 10 years from now. Market price of $1,500, implies a yield of 3.91% (semi-annual compounding); for B=$1,000 we obviously find R=10%. 28

15 Bond Yields and Prices The case of coupon bonds Coupon bonds can be represented as portfolios of zero-coupon bonds Implication for price sensitivity Suppose you purchase the 9% U.S. Treasury bond described earlier and immediately thereafter interest rates fall: APR on the bond is now 8%, compounded semiannually. What is the bond s new market price? Suppose the interest rate rises, so that the new interest rate is 12% compounded semiannually. What is the market price now? Suppose the interest equals the coupon rate of 9%. What do you observe? What are the pricing implications of these scenarios? 29 Valuing Coupon Bonds (cont.) New Semiannual interest rate = 8%/2 = 4% N c F B 1 = 1 + i i 1+ What is the price of the bond if the APR is 8% compounded semiannually? 1 B = + = , 000 * 45 $ Similarly: If R=12%: B=$ If R= 9%: B=$1, ( 1+ i) N 30

16 Relationship Between Bond Prices and Interest Rates Bond prices are inversely related to interest rates (or yields). A bond sells at par only if its interest rate equals the coupon rate A bond sells at a premium if its coupon rate is above the interest rate. A bond sells at a discount if its coupon rate is below the interest rate. 31 Interest Rate Sensitivity of Coupon Bonds Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years. The APR is 8% What are the responses to a.01% (1bp) interest rate change? Yield 5-Year Bond $ Change % Change 10-Year Bond $ Change % Change 7.995% $1, $ % $1, $ % 8.000% $1, $1, % $1, ($ ) % $1, ($ ) % DV01 $ $ Does the sensitivity of a coupon bond always increase with the term to maturity? 32

17 Bond Prices and Interest Rates $2, $2, Year Bond 10-Year Bond $1, $1, $ $0.00 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% Longer term bonds are more sensitive to changes in interest rates than shorter term bonds (holding constant the bond cashflows). 33 Bond Yields and Prices Consider the following two bonds: Both have a maturity of 5 years Both have yield of 8% First has 6% coupon, other has 10% coupon, compounded annually. Then, what are the price sensitivities of these bonds, measured by DV01 as for zero coupon bonds? Yield 6%-Bond $ Change % change 10%-Bond $ Change % change 7.995% $ $ $1, $ % $ $1, % $ ($0.1891) $1, ($0.2101) % % DV01 $ $ Why do we get different answers for two bonds with the same yield and same maturity? 34

18 Maturity and Price Risk Zero coupon bonds have well-defined relationship between maturity and interest rate sensitivity: DV01 is direct function of maturity t. Coupon bonds can have different volatilities for the same maturity DV01 now depends on maturity and coupon rate. Get different results for 6% coupon and 10% coupon bonds with same maturity. Need concept of average maturity of coupon bond: Duration 35 Duration The logical way to measure sensitivity of the bond price to changes in interest rates is to take the derivative of the price B with respect to effective rate i (see slide 22): B = i N n= 1 n c n1 N 1 ( 1+ i) + N F (1 + i) We adjust the derivative by dividing by minus the bond price and the number of periods per year m, and multiply by one plus the effective rate. The measure obtained is often called Macaulay Duration. 36

19 Duration (cont.) If we also replace n/m with T n -- which will be the time (in years) until the nth cash flow, the formula is: N (1 + i) B 1 Duration = = Tn c N 1 m B i B n ( 1+ i) + T F ( + i) N Duration is a weighted average term to maturity where the cash flows are in terms of their present value. We can rewrite the above equation in a simpler format: n= 1 PV( c ) PV( c ) PV( c ) N PV( F) Duration = T 1 + T 2 + L+ T + T 1 B 2 B N B N B 37 Duration (cont.) The duration of a bond is less than its time to maturity (except for zero coupon bonds). The duration of the bond decreases the greater the coupon rate. This is because more weight (present value weight) is being given to the coupon payments. As market interest rate increases, the duration of the bond decreases. This is a direct result of discounting. Discounting at a higher rate means lower weight on payments in the far future. Hence, the weighting of the cash flows will be more heavily placed on the early cash flows -- decreasing the duration. Modified Duration = Duration / (1+yield) 38

20 Spot Rates A spot rate is a rate agreed upon today, for a loan that is to be made today r 1 =5% indicates that the current rate for a one-year loan is 5%. r 2 =6% indicates that the current rate for a two-year loan is 6%. Etc. The term structure of interest rates is the series of spot rates r 1, r 2, r 3, We can build using STRIPS or coupon bond yields. Explanations of the term structure. Yield The Term Structure of Interest Rates An Example Maturity 40

21 Term Structure, July 1 st Term Structure, September 12 th

22 Term Structure, May 20 th,

23 45 Summary Bonds can be valued by discounting their future cash flows Bond prices change inversely with yield Price response of bond to interest rates depends on term to maturity. Works well for zero-coupon bond, but not for coupon bonds Measure interest rate sensitivity using DV01 and duration. The term structure implies terms for future borrowing: Forward rates Compare with expected future spot rates (Appendix). 46

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